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The following differential equation allows me to calculate the the number concentration of aerosol particles from 2 nm to 3 nm.

$$\frac{dN_2{_-{_3}}}{dt}=J_2-J_3-N_2{_-{_3}}.CoagS_2{_-{_3}}$$

In the above equation, $\frac{dN_2{_-{_3}}}{dt}$ is the rate of change of number concentration of the aerosol particles in $cm^-{^3}s^-{^1}$, $J_2$ is the formation rate of atmospheric particles at 2 nm in $cm^-{^3}s^-{^1}$, $J_3$ is the formation rate of atmospheric particles at 3 nm in $cm^-{^3}s^-{^1}$, $N_2{_-{_3}}$ is the number concentration of particles in $cm^-{^3}$ and $CoagS_2{_-{_3}}$ is the coagulstion sink of the aerosol particles from 2-3 nm in $s^-{^1}$.

If we assume a pseudo steady-state between the sources and sinks of 2-3 nm atmospheric molecular clusters, then: $$\frac{dN_2{_-{_3}}}{dt}=0$$

A consequence of this is:

$$N_2{_-{_3}}=\frac{J_2-J_3-N_2{_-{_3}}}{CoagS_2{_-{_3}}}$$

Number concentration of aerosol particles due to primary emissions are known to increase at larger aerosol particle sizes. Therefore, (Kulmala et al., 2021) states that by using a higher value of 25 nm instead of 3 nm as the upper interval bound $d_2$, which is the upper interval bound aerosol particle diameter in nm, for the calculation of $J_2$ and $J_3$, the influence of primary emissions on $N_2{_-{_3}}$ are reduced. The apparent logic of taking a larger size interval for the calculation of $J_2{_-{_3}}$ is that the increase in the primary emissions at larger sizes are offsetting the formation rate at both the lower interval bound $d_1$ and the upper interval bound $d_2$. Although, in terms of absolute numbers, the calculated values of $J_2$ and $J_3$ would be less accurate using this larger size interval. Mathematically:

$$N_2{_-{_3}}=N_2{_-{_2{_5}}}-N_3{_-{_2{_5}}}$$

Could someone explain to me, mathematically, how is choosing a higher value of 25 nm instead of 3 nm as the upper interval bound $d_2$ to calculate $N_2{_-{_3}}$, offsetting the formation rate at both the lower interval bound $d_1$, 2 nm, and the upper interval bound $d_2$, 3 nm?

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